James F. Glazebrook, Marcos Jardim, Franz W. Kamber
A Fourier-Mukai Transform for Real Torus Bundles
Preprint series: ESI preprints

MSC:

65R10 Integral transforms

53C12 Foliations (differential geometric aspects), See Also {57R30, 57R32}

14D20 Algebraic moduli problems, moduli of vector bundles, {For analytic moduli problems, See 32G13}

Abstract: We construct a Fourier--Mukai transform for smooth complex vector
bundles $E$ over a torus bundle $\pi:M \to B~,$ the vector bundles
being endowed with various structures of increasing complexity.
At a minimum, we consider vector bundles $E$ with a flat partial
unitary connection, that is families or deformations of flat
vector bundles (or unitary local systems) on the torus $T~.$
This leads to a correspondence between such objects on $M$ and
relative skyscraper sheaves $\cS$ supported on a spectral covering
$\Sigma \hra \what M~,$ where $\hat\pi:\what{M} \to B$ is the flat
dual fiber bundle. Additional structures on $(E,\nabla)$ (flatness, anti-self-duality)
will be reflected by corresponding data on the transform $(\cS, \Sigma)~.$
Several variations of this construction will be presented,
emphasizing the aspects of foliation theory which enter into
this picture.

Keywords: Fourier-Mukai transforms, foliation theory, unitary local systems, instantons, monopoles